Lets say you go and hire a C150 from your local flying club as a licensed pilot; so far as the CAA is concerned this is a private flight from an operational viewpoint, but a public transport flight from an airworthiness viewpoint. Hence you only need a PPL to hire an aeroplane, but it still needs a PT CofA.
Where instruction is concerned, there is a similar division. The flight is considered public transport from an airworthiness viewpoint (the AIC exempts you from this rule, but doesn't actually change it where sole-owner training is concerned) and also from an operational viewpoint, which is why the instructor needs a commercial license.
So from the operational viewpoint, so long as the instructor was properly licensed and you used a licensed airfield, the flight would be operationally legitimate and you'd almost certainly get your license / rating from CAA Personnel Licensing. But from an airworthiness viewpoint the flight would be illegal because it was conducted without holding the relevant CofA on the aeroplane. So, whilst one bit of the CAA (PLD) would probably give you your license, another bit (aircraft certification and/or legal and enforcement) would be prosecuting your instructor for breaching the ANO with respect to the airworthiness regulations.
G